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Theorem cdlemkuat 29959
Description: Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
Hypotheses
Ref Expression
cdlemk1.b  |-  B  =  ( Base `  K
)
cdlemk1.l  |-  .<_  =  ( le `  K )
cdlemk1.j  |-  .\/  =  ( join `  K )
cdlemk1.m  |-  ./\  =  ( meet `  K )
cdlemk1.a  |-  A  =  ( Atoms `  K )
cdlemk1.h  |-  H  =  ( LHyp `  K
)
cdlemk1.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemk1.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemk1.s  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
cdlemk1.o  |-  O  =  ( S `  D
)
cdlemk1.u  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
Assertion
Ref Expression
cdlemkuat  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  e.  A
)
Distinct variable groups:    f, i,  ./\    .<_ , i    .\/ , f, i    A, i    D, f, i    f, F, i    i, H    i, K    f, N, i    P, f, i    R, f, i    T, f, i    f, W, i    ./\ , e    .\/ , e    D, e, j    e, G, j   
e, O    P, e    R, e    T, e    e, W    ./\ , j    .<_ , j    .\/ , j    A, j    D, j    j, F   
j, H    j, K    j, N    j, O    P, j    R, j    T, j   
j, W
Allowed substitution hints:    A( e, f)    B( e, f, i, j)    S( e, f, i, j)    U( e, f, i, j)    F( e)    G( f, i)    H( e, f)    K( e, f)    .<_ ( e, f)    N( e)    O( f, i)

Proof of Theorem cdlemkuat
StepHypRef Expression
1 simp11 990 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 cdlemk1.b . . 3  |-  B  =  ( Base `  K
)
3 cdlemk1.l . . 3  |-  .<_  =  ( le `  K )
4 cdlemk1.j . . 3  |-  .\/  =  ( join `  K )
5 cdlemk1.m . . 3  |-  ./\  =  ( meet `  K )
6 cdlemk1.a . . 3  |-  A  =  ( Atoms `  K )
7 cdlemk1.h . . 3  |-  H  =  ( LHyp `  K
)
8 cdlemk1.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
9 cdlemk1.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
10 cdlemk1.s . . 3  |-  S  =  ( f  e.  T  |->  ( iota_ i  e.  T
( i `  P
)  =  ( ( P  .\/  ( R `
 f ) ) 
./\  ( ( N `
 P )  .\/  ( R `  ( f  o.  `' F ) ) ) ) ) )
11 cdlemk1.o . . 3  |-  O  =  ( S `  D
)
12 cdlemk1.u . . 3  |-  U  =  ( e  e.  T  |->  ( iota_ j  e.  T
( j `  P
)  =  ( ( P  .\/  ( R `
 e ) ) 
./\  ( ( O `
 P )  .\/  ( R `  ( e  o.  `' D ) ) ) ) ) )
132, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdlemkuel 29958 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( U `  G )  e.  T
)
14 simp33l 1087 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  P  e.  A )
153, 6, 7, 8ltrnat 29233 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  G )  e.  T  /\  P  e.  A
)  ->  ( ( U `  G ) `  P )  e.  A
)
161, 13, 14, 15syl3anc 1187 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( R `  F )  =  ( R `  N )  /\  G  e.  T )  /\  ( F  e.  T  /\  D  e.  T  /\  N  e.  T )  /\  ( ( ( R `
 D )  =/=  ( R `  F
)  /\  ( R `  D )  =/=  ( R `  G )
)  /\  ( F  =/=  (  _I  |`  B )  /\  G  =/=  (  _I  |`  B )  /\  D  =/=  (  _I  |`  B ) )  /\  ( P  e.  A  /\  -.  P  .<_  W ) ) )  ->  ( ( U `  G ) `  P )  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920    e. cmpt 3974    _I cid 4197   `'ccnv 4579    |` cres 4582    o. ccom 4584   ` cfv 4592  (class class class)co 5710   iota_crio 6181   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   trLctrl 29251
This theorem is referenced by:  cdlemk7u  29963  cdlemk11u  29964  cdlemk12u  29965
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252
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